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Theorem stoweidlem22 29815
Description: If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem22.8  |-  F/ t
ph
stoweidlem22.9  |-  F/_ t F
stoweidlem22.10  |-  F/_ t G
stoweidlem22.1  |-  H  =  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t )
) )
stoweidlem22.2  |-  I  =  ( t  e.  T  |-> 
-u 1 )
stoweidlem22.3  |-  L  =  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t )
) )
stoweidlem22.4  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
stoweidlem22.5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem22.6  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
stoweidlem22.7  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
Assertion
Ref Expression
stoweidlem22  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
Distinct variable groups:    f, g,
t, A    f, F, g    f, G, g    f, I, g    T, f, g, t    ph, f, g    g, L    x, t, A    x, T    ph, x
Allowed substitution hints:    ph( t)    F( x, t)    G( x, t)    H( x, t, f, g)    I( x, t)    L( x, t, f)

Proof of Theorem stoweidlem22
StepHypRef Expression
1 stoweidlem22.8 . . . 4  |-  F/ t
ph
2 stoweidlem22.9 . . . . 5  |-  F/_ t F
32nfel1 2588 . . . 4  |-  F/ t  F  e.  A
4 stoweidlem22.10 . . . . 5  |-  F/_ t G
54nfel1 2588 . . . 4  |-  F/ t  G  e.  A
61, 3, 5nf3an 1863 . . 3  |-  F/ t ( ph  /\  F  e.  A  /\  G  e.  A )
7 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  t  e.  T )
8 simpl1 991 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ph )
9 stoweidlem22.2 . . . . . . . . . . . 12  |-  I  =  ( t  e.  T  |-> 
-u 1 )
10 neg1rr 10425 . . . . . . . . . . . . 13  |-  -u 1  e.  RR
11 stoweidlem22.7 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
1211stoweidlem4 29797 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -u 1  e.  RR )  ->  (
t  e.  T  |->  -u
1 )  e.  A
)
1310, 12mpan2 671 . . . . . . . . . . . 12  |-  ( ph  ->  ( t  e.  T  |-> 
-u 1 )  e.  A )
149, 13syl5eqel 2526 . . . . . . . . . . 11  |-  ( ph  ->  I  e.  A )
15 eleq1 2502 . . . . . . . . . . . . . . 15  |-  ( f  =  I  ->  (
f  e.  A  <->  I  e.  A ) )
1615anbi2d 703 . . . . . . . . . . . . . 14  |-  ( f  =  I  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  I  e.  A ) ) )
17 feq1 5541 . . . . . . . . . . . . . 14  |-  ( f  =  I  ->  (
f : T --> RR  <->  I : T
--> RR ) )
1816, 17imbi12d 320 . . . . . . . . . . . . 13  |-  ( f  =  I  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  I  e.  A )  ->  I : T --> RR ) ) )
19 stoweidlem22.4 . . . . . . . . . . . . 13  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
2018, 19vtoclg 3029 . . . . . . . . . . . 12  |-  ( I  e.  A  ->  (
( ph  /\  I  e.  A )  ->  I : T --> RR ) )
2120anabsi7 815 . . . . . . . . . . 11  |-  ( (
ph  /\  I  e.  A )  ->  I : T --> RR )
2214, 21mpdan 668 . . . . . . . . . 10  |-  ( ph  ->  I : T --> RR )
238, 22syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  I : T --> RR )
2423, 7ffvelrnd 5843 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
I `  t )  e.  RR )
25 simpl3 993 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  G  e.  A )
26 eleq1 2502 . . . . . . . . . . . . . . 15  |-  ( f  =  G  ->  (
f  e.  A  <->  G  e.  A ) )
2726anbi2d 703 . . . . . . . . . . . . . 14  |-  ( f  =  G  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  G  e.  A ) ) )
28 feq1 5541 . . . . . . . . . . . . . 14  |-  ( f  =  G  ->  (
f : T --> RR  <->  G : T
--> RR ) )
2927, 28imbi12d 320 . . . . . . . . . . . . 13  |-  ( f  =  G  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  G  e.  A )  ->  G : T --> RR ) ) )
3029, 19vtoclg 3029 . . . . . . . . . . . 12  |-  ( G  e.  A  ->  (
( ph  /\  G  e.  A )  ->  G : T --> RR ) )
3130anabsi7 815 . . . . . . . . . . 11  |-  ( (
ph  /\  G  e.  A )  ->  G : T --> RR )
32313adant3 1008 . . . . . . . . . 10  |-  ( (
ph  /\  G  e.  A  /\  t  e.  T
)  ->  G : T
--> RR )
33 simp3 990 . . . . . . . . . 10  |-  ( (
ph  /\  G  e.  A  /\  t  e.  T
)  ->  t  e.  T )
3432, 33ffvelrnd 5843 . . . . . . . . 9  |-  ( (
ph  /\  G  e.  A  /\  t  e.  T
)  ->  ( G `  t )  e.  RR )
358, 25, 7, 34syl3anc 1218 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( G `  t )  e.  RR )
3624, 35remulcld 9413 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( I `  t
)  x.  ( G `
 t ) )  e.  RR )
37 stoweidlem22.3 . . . . . . . 8  |-  L  =  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t )
) )
3837fvmpt2 5780 . . . . . . 7  |-  ( ( t  e.  T  /\  ( ( I `  t )  x.  ( G `  t )
)  e.  RR )  ->  ( L `  t )  =  ( ( I `  t
)  x.  ( G `
 t ) ) )
397, 36, 38syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( L `  t )  =  ( ( I `
 t )  x.  ( G `  t
) ) )
409fvmpt2 5780 . . . . . . . . 9  |-  ( ( t  e.  T  /\  -u 1  e.  RR )  ->  ( I `  t )  =  -u
1 )
4110, 40mpan2 671 . . . . . . . 8  |-  ( t  e.  T  ->  (
I `  t )  =  -u 1 )
4241adantl 466 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
I `  t )  =  -u 1 )
4342oveq1d 6105 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( I `  t
)  x.  ( G `
 t ) )  =  ( -u 1  x.  ( G `  t
) ) )
4435recnd 9411 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( G `  t )  e.  CC )
4544mulm1d 9795 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( -u 1  x.  ( G `
 t ) )  =  -u ( G `  t ) )
4639, 43, 453eqtrd 2478 . . . . 5  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( L `  t )  =  -u ( G `  t ) )
4746oveq2d 6106 . . . 4  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( F `  t
)  +  ( L `
 t ) )  =  ( ( F `
 t )  + 
-u ( G `  t ) ) )
48 simpl2 992 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  F  e.  A )
49 eleq1 2502 . . . . . . . . . . . 12  |-  ( f  =  F  ->  (
f  e.  A  <->  F  e.  A ) )
5049anbi2d 703 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  F  e.  A ) ) )
51 feq1 5541 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f : T --> RR  <->  F : T
--> RR ) )
5250, 51imbi12d 320 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  F  e.  A )  ->  F : T --> RR ) ) )
5352, 19vtoclg 3029 . . . . . . . . 9  |-  ( F  e.  A  ->  (
( ph  /\  F  e.  A )  ->  F : T --> RR ) )
5453anabsi7 815 . . . . . . . 8  |-  ( (
ph  /\  F  e.  A )  ->  F : T --> RR )
558, 48, 54syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  F : T --> RR )
5655, 7ffvelrnd 5843 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
5756recnd 9411 . . . . 5  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
5857, 44negsubd 9724 . . . 4  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( F `  t
)  +  -u ( G `  t )
)  =  ( ( F `  t )  -  ( G `  t ) ) )
5947, 58eqtr2d 2475 . . 3  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( F `  t
)  -  ( G `
 t ) )  =  ( ( F `
 t )  +  ( L `  t
) ) )
606, 59mpteq2da 4376 . 2  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  =  ( t  e.  T  |->  ( ( F `
 t )  +  ( L `  t
) ) ) )
61143ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  I  e.  A )
62 nfmpt1 4380 . . . . . . . 8  |-  F/_ t
( t  e.  T  |-> 
-u 1 )
639, 62nfcxfr 2575 . . . . . . 7  |-  F/_ t
I
6463nfeq2 2589 . . . . . 6  |-  F/ t  f  =  I
654nfeq2 2589 . . . . . 6  |-  F/ t  g  =  G
66 stoweidlem22.6 . . . . . 6  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
6764, 65, 66stoweidlem6 29799 . . . . 5  |-  ( (
ph  /\  I  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t ) ) )  e.  A )
6861, 67syld3an2 1265 . . . 4  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t ) ) )  e.  A )
6937, 68syl5eqel 2526 . . 3  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  L  e.  A )
70 stoweidlem22.5 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
71 nfmpt1 4380 . . . . 5  |-  F/_ t
( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t )
) )
7237, 71nfcxfr 2575 . . . 4  |-  F/_ t L
7370, 2, 72stoweidlem8 29801 . . 3  |-  ( (
ph  /\  F  e.  A  /\  L  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( L `  t ) ) )  e.  A )
7469, 73syld3an3 1263 . 2  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( L `  t ) ) )  e.  A )
7560, 74eqeltrd 2516 1  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   F/wnf 1589    e. wcel 1756   F/_wnfc 2565    e. cmpt 4349   -->wf 5413   ` cfv 5417  (class class class)co 6090   RRcr 9280   1c1 9282    + caddc 9284    x. cmul 9286    - cmin 9594   -ucneg 9595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-ltxr 9422  df-sub 9596  df-neg 9597
This theorem is referenced by:  stoweidlem33  29826
  Copyright terms: Public domain W3C validator